Yup, You can plug the new wireless router into an exsisting open port on the wired router or you could get rid of your wired router and replce it with the wireless, plugging whatever is in the old router now into the wireless routers wired ports.

This is true if the current router you are using is a modem/router combo AND you dont want to use it as a router anymore. Then you set it to bridge (passthorugh modem) and use the wireless routers wired ports.

This is not nessesary. You could simply plug the new wireless router into an open port on the exsisting router and use it that way.

If the modem is a modem/router, for your wireless router to get online, make sure to check the IP address of both modem/router and your wireless router. and also, make sure the modem/ router is not in bridged mode.
that should work...

i bought a wireless router not knowing how it would work. my original modem had a usb to pc and ethernet to pc, and the pc was working thru the ethernet. now i set it up so the pc uses the usb, i set the wireless router thru the ethernet and (french word, voila or something like that)